In AIR, Misuzu dreams of the time of Kannabi-no-Mikoto which was her "full-self". As the dream becomes more vivid, her body deteriorates because no human body can contain the entirety of Kanna's soul, eventually leading to her tragic death.
In Cardcaptor Sakura, the eponymous character dreams of the time of death not of Clow Reed.
Happens several time to different characters in Shaman King.
A different take happens between Koyuki and Snow in MÄR, they are essentially the counterparts of the same person, and so they share dreams of what's happening between their worlds. Snow dreams of Koyuki's memories of her and Ginta's past, while Koyuki dreams of Snow's interactions with Ginta, and relates them to his worried mother to confirm to her that her son is alright, though in another world. Given that both girls love Ginta, this level of interaction helps to solidify them as his 'One True Love', yes, both of them.
An old man's astral form entered Ranma's, from Ranma ½, dream showing him a date that had happened when the man was young.
Frodo dreams of Gandalf's capture and escape from Isengard during Fellowship Of The Ring. Gandalf observed "Then it was late in coming". (Worth noting since he also has prophetic dreams.)
Merry, under the barrow wight's spell, dreamed of falling in battle after a night attack - evidently of the men whose grave they were in.
Faramir often dreams of the fall of Númenor, which happened an age ago.
In Terry Pratchett and Neil Gaiman's Good Omens, Witchfinder Shadwell has a dream about seventeenth-century witch and prophet Agnes Nutter. Because of her prophetic ability, he is detected by her, leading to a Brick Joke; in the first account the book gives of her burning at the stake, it says that her last act was to look up at the sky and say "That goes for you as well, you daft old fool," and it's implied that she's talking to God. When Shadwell dreams about it, it becomes clear that she was actually saying it to him.
Happens to Aya Brea in Parasite Eve 2. It's not so much sleeping as it's fainting from exhaustion.
A major plot point in Final Fantasy VIII, and several gameplay sequences, are devoted to the main characters dreaming of past events involving Laguna Loire and his friends. This later turns out to be because of another character's magical power.
In Xenogears, Fei and Elly experience visions of themselves in the distant past. It's not until near the end of the game that the reason for these visions is revealed.
In Forever's End, Epoch regularly dreams of the past of the Crusades.
In the Girls LoveVisual NovelAkai Ito, Kei dreams of the time of the Mizuki Tribe and their coalition with the human exorcists from the Capitol in their quest to vanquish Nushi. She also dreams of the fall of the Mizuki at the hands of their human ex-allies, and the subsequent suffering of Sakuya.
Its sort-of sequel, Aoi Shiro, Syouko dreams of the time of Yasuhime and her exile to the southern islands to seal the <<Sword>>. This is Justified by having the blood of Yasuhime literally flowing in her veins — she was brought back to life by Yasuhime, once.
In Homestuck, an important Troll historical figure known as the Signless had dreams of life before he Scratched his universe, causing everyone's lives to be reset, inspiring him to spread anti-caste-system messages.
In UC, all four main characters have repeated shared dreams of a ritual involving a candle and a mysterious figure yelling “What have you done!” There is no in comic explanation about these dreams yet, but it is strongly hinted that they are showing a ritual that occurred before the story began.
Parodied in the South Park episode "I'm a Little Bit Country", in which Cartman tries to invoke this trope to get out of studying for an American history report.
In The Legend Of Korra, Avatar Korra occasionally has visions of her past life as Avatar Aang that are her past incarnation's warnings about Tarrlok's ambition and his bloodbending, both of which he inherited from his father Yakone. They were also possibly warnings of Tarrlok's brother Noatak, who was later on revealed to be Amon.
In the season four premiere of My Little Pony: Friendship is Magic, Twilight Sparkle drinks a potion that responds only to alicorn magic in order to find out why the Everfree Forest is invading and figure out where the missing Celestia and Luna are. The potion causes the drinker to view events that have long since passed. Twilight witnesses the banishment of Nightmare Moon, the defeat of Discord by Celestia and Luna, and how the Elements of Harmony came to be in the possession of the two royal pony sisters.